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Eprints ID : 15724
To link to this article : DOI:10.1016/j.anucene.2016.04.007
URL : http://dx.doi.org/10.1016/j.anucene.2016.04.007
To cite this version : Chikhi, Nourdine and Clavier, Rémi and
Laurent, Jean-Paul and Fichot, Florian and Quintard, Michel Pressure
drop and average void fraction measurements for two-phase flow
through highly permeable porous media. (2016) Annals of Nuclear
Energy, vol. 94. pp. 422 - 432. ISSN 0306-4549
Any correspondence concerning this service should be sent to the repository
administrator: [email protected]
Pressure drop and average void fraction measurements for two-phase
flow through highly permeable porous media
N. Chikhi a,⇑, R. Clavier a, J.-P. Laurent b,c, F. Fichot d, M. Quintard e,f
a
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SEREX/LE2M, Cadarache bât. 327, 13115 St Paul-lez-Durance, France
Univ. Grenoble Alpes, LTHE, F-38000 Grenoble, France
c
CNRS, LTHE, F-38000 Grenoble, France
d
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SAG/LEPC, Cadarache bât. 700, 13115 St Paul-lez-Durance, France
e
Université de Toulouse, INPT, UPS, Institut de Mécanique des Fluides de Toulouse (IMFT), Allée Camille Soula, F-31400 Toulouse, France
f
CNRS, IMFT, F-31400 Toulouse, France
b
a b s t r a c t
Keywords:
Debris bed
Pressure drop
Void fraction
Interfacial drag
The modeling of pressure drop for two-phase flows through porous media is a key point to assess the
coolability of debris beds resulting from nuclear severe accidents. Models involve several parameters
which are non-linear functions of the void fraction, e.g. relative permeabilities. Their identification
requires that experimental data include the measurement of void fraction. This paper presents a new
technique developed to reach this objective. The method is based on the use of a capacitance probe
and has been validated by comparison with a weighing method. The validation has shown that the
accuracy is better than 10%. The measurement device has been implemented in the CALIDE facility, at
IRSN, which has been designed to perform air–water flow through debris bed. Tests have been carried
out with beds made of single size 4 mm and 8 mm beads. Measurements of pressure drop and average
void fraction are reported in the paper, for air and water flow rates representative of flows that would
result of either the reflooding of the damaged core or the cooling of corium debris in a stagnant pool
of water. Finally, the pressure drop models used in severe accident simulation codes, based on generalizations of the single-phase Ergun law, have been assessed against the new data. It has been observed
that generalized Ergun laws including an interfacial drag term accurately predict the pressure drop
and the void fraction for flows with a zero net water velocity.
1. Introduction
The understanding of two-phase flow through high permeability porous media is of interest in several applications such as
chemical reactors, oil/gas production or soil physics. In the field
of nuclear safety analysis, the study of such flows has become of
great importance after the TMI-2 accident (1979), where the injection of water on hot debris bed have stopped the progression of
melting and led to keep the corium inside the reactor vessel
(Broughton et al., 1989). By now, the assessment of debris coolability remains an open question. To address it, there is a need for a
reliable thermal–hydraulic model able to estimate the debris heat
removal rate during quenching or dry-out.
Several experimental investigations have been carried out to
help modeling. Dry-out experiments (Hardee and Nilson, 1977;
⇑ Corresponding author.
E-mail address: [email protected] (N. Chikhi).
Dhir and Catton, 1977; Lipinski, 1984; Decossin, 1999) and reflood
tests (top flooding: (Cho et al., 1984; Ginsberg et al., 1982; Tutu
et al., 1984a; Tung and Dhir, 1988); bottom flooding: (Hall and
Hall, 1981; Tutu et al., 1984c; Tung and Dhir, 1986)) have been performed, providing global parameters: outlet steam flow rate,
quench front velocity, etc. The instrumentation was scarce so that
only 0D/1D models have been developed, based on the counter
current flow limitation (CCFL) and on correlations for the friction
laws (Lipinski, 1982; Reed, 1982; Hu and Theofanous, 1991; Tung
and Dhir, 1988; Schulenberg and Müller, 1987). 2D/3D models,
which provide detailed flow fields (Fichot et al., 2006; Burger
et al., 2006), have been developed. However, due to the lack of
accurate local measurements in debris beds, models could not be
validated. It is worth noticing that new data are available on large
scale reflood tests (Chikhi et al., 2015). Two kinds of closure laws
are required in these models: friction laws and heat transfer laws.
Since the generalization of Darcy’s law by Muskat (1937), pressure drop models for two-phase flow through porous media
involves parameters non-linearly dependent on saturation. In the
context of nuclear safety, it is generally assumed (Schmidt, 2004)
that the pressure drop is a function of the superficial gas and liquid
velocities and of the void fraction (or gas saturation), i.e.,
DP ¼ f ðV l ; V g ; aÞ. Therefore, the inclusion of void fraction measurements in experimental data would seem an obvious requirement.
And yet, the only data available in this field come from the experiments of Tutu et al. (1984b) with a zero net water flow. Schmidt
(2004) compared the friction laws with these experimental data
and showed that the pressure drop models should include an explicit interfacial friction term. Tung and Dhir proposed a modification
of their model to take into account this feature by introducing a
flow regime map (Tung and Dhir, 1988). Nevertheless, due to the
lack of experimental data, the flow regime map could not be
validated.
The bibliography related to the transposition of air–water flow
to steam–water flow through porous media is quite scarce in the
nuclear field. By now, there is no definitive conclusion on in this
subject. Some studies have been done in petroleum and geothermal engineering field and a large review of results on steam–water
relative permeability (laminar regime) can be found in Counsil
(1979): Corey’s equations to characterize steam–water relative
permeability (Corey, 1977), Chen’s drainage relative permeability
curves (Chen et al., 1978) and Horne’s curves relative permeability
curves (Horne and Ramey, 1978). Few attentions have been paid to
steam–water relative passability (inertial regime). More recently,
discrepancies between steam and nitrogen injection have been
reported in Jabbour et al. (1996).
The aim of the present paper is to propose an original method to
measure the void fraction for air–water flows through nuclear-like
debris bed. The method is based on the use of capacitance probes,
widely used in soil physics (Zakri et al., 1998). It is presented
in detail in Section 3. The measurement set-up has been
implemented in the CALIDE facility (Chikhi et al., 2013), which is
an air–water loop designed to generate two-phase flows through
porous media. The CALIDE facility and the test conditions (flow
regime and debris bed characteristics) are presented in Section 2.
The capacitance probe method has been validated by comparison
with a weighing method (Section 3). Different kinds of particles
have been used to make the porous beds: glass beads, ceramic
prisms and ceramic cylinders. Finally, pressure drop measurements
have been performed on beds made of single-size spherical particles. Two beds have been built with 4 mm and 8 mm diameter
spheres. The pressure drop have been measured for several gas
and liquid mass flow rates that are representative of severe accident
conditions. Section 4 is devoted to the presentation of these original
experimental data. In Section 5, the classical pressure drop models
cited above (Lipinski, Reed, Hu and Theofeanous, Tung and Dhir,
Schulenberger and Muller), which are implemented in severe
accident codes, are assessed against the experimental results.
The fluid flow rates are measured and controlled by five high
precision BronkhorstÒ flowmeters with specific measuring ranges
(see Table 1). Six radial holes uniformly distributed along the test
section allow pressure tapping at different levels inside,
downstream and upstream the debris bed. Pressure drops are
measured by two Rosemount-3051Ò differential pressure sensors
(see Table 2). A thermocouple records the temperature at the top
of the test section. This measured temperature is used for fluid
viscosity l and density q calculation using tabulated values
(Lide, 1990). The density is calculated for the average pressure to
account for the gas compressibility (see Clavier et al. (2015) for
more details).
2.2. Debris bed
A comprehensive review of the geometrical characteristics of
nuclear debris beds can be found in Chikhi et al. (2014). One of
the main conclusions of this study was that the particle size ranges
from 0.3 to 10 mm. The porosity ranges from 0:35 to 0:55. It was
also shown that, as far as pressure drop estimation is concerned,
a debris bed made of polydisperse and non-spherical particles
can be represented by an equivalent bed made of single-sized
spherical particles either in the study of quenching or dry-out
situation.
To validate the void fraction measurement method, a bed made
of particles with identical shapes has been used. Several shapes were
tested: spheres (glass), prisms (ceramic) and cylinders (ceramic)
(Fig. 2). Their geometrical characteristics are given in Table 3.
For the integral tests devoted to pressure drop measurements,
two beds have been studied, made of 4 mm and 8 mm glass beads.
The bed properties have been measured and are given in Table 4.
Bed porosities have been measured as the pressure losses are
very sensitive to this parameter. To determine the bed porosity,
the test section was filled with water with a mass mw . The porosity
is deduced using the water density qw :
e¼
4mw
qw pD2 H
;
ð1Þ
where H is the test section height and D the test section diameter.
According to the particle diameter and to the porosity, two
other bed characteristics can be calculated, the permeability K
and the passability g:
K¼
g¼
e3 d2
hK ð1 eÞ2
e3 d
hg ð1 eÞ
;
;
ð2Þ
ð3Þ
2. Experimental set-up
where hK ¼ 181 and hg ¼ 1:63 are two empirical constants, which
have been determined by fitting Ergun’s law (Ergun, 1952)
2.1. The CALIDE loop
The CALIDE facility is an air/water single- and two-phase flow
loop designed for pressure drop vs flow rate measurements for
flow through porous media (Clavier et al. (2015) and Fig. 1). The
test section containing the bed is made of a Plexiglas pipe
(500 mm high and 94 mm diameter) which allows flow visualization. Air is supplied from the bottom and flows up through the
bed, while water can flood the bed either from the top or the
bottom, providing either co-current or counter-current flows.
A stainless steel wire mesh is placed at the bottom of the test
section to support the bed. This wire mesh has a negligible impact
on the total pressure drop measured across the test section.
@P
l
q
þ qg ¼ U þ U 2 ;
g
@z
K
ð4Þ
against experiments as presented in a previous paper (Clavier et al.,
2015) in the case of one-phase flows through beds packed with
single-sized beads.
2.3. Flow regime and test conduct
The water and air flow rates have been chosen to be representative of water and steam filtration velocities during the reflood
of a damaged core. Considering a reflood at low pressure in a
French 1300 MWe PWR, the reflood mass flow rate is equivalent
to a filtration velocity ranging from 5 mm/s to 32 mm/s. The steam
Air outlet
Acquisition
device
TCK
Pdif
Pdif
Pabs
Water
tank
Fig. 1. The CALIDE facility.
Table 1
CALIDE flow meters.
Flow meters
BRONKHORSTÒ
BRONKHORSTÒ
BRONKHORSTÒ
BRONKHORSTÒ
BRONKHORSTÒ
F-201-CV
F-202-CV
F-203-CV
Cori-Flow M14
Cori-Flow M15
Type
Range
Accuracy
Air Flow
Air Flow
Air Flow
Water Flow
Water Flow
0–10 NL/min
0–200 NL/min
0–1000 NL/min
0–30 kg/h
0–600 kg/h
0.5%
0.1%
0.1%
0.2%
0.2%
Reading
Reading
Reading
Reading
Reading
Table 2
CALIDE pressure sensors.
Pressure sensors
Type
Range
Max. range (MR)
Accuracy
ROSEMOUNTÒ3051
ROSEMOUNTÒ3051
ROSEMOUNTÒ3051
Differential
Differential
Absolute
7; +7 mbar
10; +200 mbar
0; 2 bar
14.19 mbar
1246 mbar
55.2 bar
0.045% MR
0.04% Reading + 0.023% MR
0.025% Range
flow rate can be deduced from debris bed reflood experiments.
According to PRELUDE tests (Repetto et al., 2013), the steam filtration velocities ranges from 1 m/s to 7 m/s.
Two-phase flows have been generated into the two beds presented in the previous paragraph and in Table 4, and pressure drop
measurements have been performed with the following conduct.
The water mass flow rate was increased step by step for a given
air mass flow rate. Each step had a duration of more than 180s
so that it can be assumed that the flow was stationary. This was
confirmed by visual observations and stationarity of all probe measurements. The main parameters necessary for modeling have
been measured: the pressure drop DP, the filtration velocities U l
and U g , and the void fraction a.
The test was also divided into four phases (see Fig. 3):
low water and air flow rate: m_ l ¼ 0 ! 30 kg/h, m_ g ¼ 0 ! 10 NL/
min,
high water flow rate, low air flow rate: m_ l ¼ 0 ! 600 kg/h,
m_ g ¼ 0 ! 10 NL/min,
low water flow rate, high air flow rate: m_ l ¼ 0 ! 30 kg/h,
m_ g ¼ 0 ! 200 NL/min,
high water and air flow rate: m_ l ¼ 0 ! 600 kg/h,
m_ g ¼ 0 ! 200 NL/min,
where m_ l and m_ g are respectively the liquid and gas mass flow rate.
Thus, the CALIDE flowmeters have a range of filtration velocity
from 0 to 24 mm/s 600 kg/h for water and from 0 to 0.5 m/
s 200 NL/min for air. The maximum air flow rate had to be
restricted to prevent fluidization of the bed.
Fig. 2. Particles used to validate the void fraction measurement.
Table 3
Bed properties – validation.
Bed name
Material
Diameter/side (mm)
Height (mm)
Density (kg/3)
Porosity
B3 – 1
B3 – 2
B4 – 1
B4 – 2
B8 – 1
B8 – 2
C5*5
C8*12–1
C8*12–2
P4*4
Glass spheres
Glass spheres
Glass spheres
Glass spheres
Glass spheres
Glass spheres
Ceramic cylinder
Ceramic cylinder
Ceramic cylinder
Ceramic prism
2.940 0.044
2.940 0.044
4.058 0.031
4.058 0.031
7.877 0.116
7.877 0.116
5.13 0.08
7.99 0.10
7.99 0.10
4.15 0.11
–
–
–
–
–
–
4.53 0.23
11.16 0.48
11.16 0.48
3.84 0.13
2560 7.4
2560 7.4
2560 7.4
2560 7.4
2560 7.4
2568 7.4
2572 7.4
2568 7.4
2568 7.4
2568 7.4
0.380
0.399
0.378
0.379
0.391
0.391
0.331
0.357
0.357
0.331
Several definitions of Reynolds numbers exist to characterize
the flow regime in porous media (Kaviany, 1995). Here, the following expression is used as it suits well the granular medium:
Re ¼
probe application can be found in Gardner et al. (1998), Council
and Ramey (1979) and Chen et al. (1978).
3.1. Capacitive probes and set-up
dU q
;
lð1 eÞ
ð5Þ
where d is the average particle size, U is the filtration velocity,
the fluid viscosity. It gives 0 < Rel < 300 and 0 < Reg < 500.
l is
3. Void fraction measurement for two-phase flow through
porous media
Considering two-phase flows through porous media, the void
fraction a is defined as the ratio between the volume occupied
by the gas and the whole volume accessible to the fluid. It is related
to the liquid ‘‘saturation”, as commonly used in porous media physics, by the relation s ¼ 1 a. Generally, the void fraction measurement in a porous medium is indirect, taking advantage of the
differences of physical properties in each phase (liquid, gas and
solid). Several techniques have been developed. Tutu et al.
(1984b) used a weighting method, which works with a zero net
water flow only and is suited to the measurement of an average
saturation. Radiation methods –infra-red (Berhold et al., 1994),
X-ray (Geffen and Gladfelter, 1952), b-ray (Zirnig, 1978), c-ray
(Yano, 1984)– were also used. They often allow for the measurement of local saturation within the porous medium, but they
require heavy facilities and safety procedures. By analogy, ultrasound wave methods have been employed (Bensler, 1990) as well
as Nuclear Magnetic Resonance (Saraf and Fatt, 1967). In our study,
a new technique based on the use of capacitance probes has been
developed, and will be presented in this section. Note that it has
been protected by a patent (FR1551375). Example of capacitance
The capacitance probe CS-616 by Campbell ScientificÒ has been
implemented into the CALIDE facility to measure the void fraction.
It consists in two metal rods (L 30 cm) connected to a sensor
head. The metal rods are inserted in the debris bed (see Fig. 4),
forming an electrical capacitor (C). This capacitor is part of a RLC
circuit, the resistor R and the inductor L being contained in the
head sensor. The head sensor also contains an electric oscillator
which is able to adjust its frequency f on the resonance frequency
of the RLC circuit ‘debris bed + rods + sensor head’. The sensor head
is linked by a waterproof connexion to an oscilloscope, thus providing the frequency f. The frequency depends on the capacitance
C, which is proportional to the dielectric permittivity of the medium in which the rods are inserted. The dielectric permittivity itself
depends on the medium composition (air/water/glass balls) and
therefore on the void fraction, as the bed is supposed to be fixed.
It means that the void fraction can be deduced from the frequency
measurement, if a relation between the dielectric permittivity and
the void fraction can be determined.
3.2. Effect of void fraction on permitivity: Lichtenecker modeling
The Lichtenecker model (Lichtenecker, 1926) proposes a
relation between the global relative dielectric permittivity K of a
mixture of N phases and their relative dielectric permittivities
and volume fractions K i and ei :
Ka ¼
X
ei K ai :
ð6Þ
i¼1;N
Table 4
Bed properties – pressure drop measurement.
Bed no
Glass spheres (mm)
Diameter (mm)
Balls density (kg/3)
Porosity
Permeability (m2)
Passability (m)
1
2
4
8
4.058 0.031
7.877 0.116
2560 7.4
2568 7.4
0.375
0.386
1.21E8
5.37E8
2.09E4
4.60E4
ml
(kg/h)
600
2
are two situations where the value of the void fraction is obvious:
first, when the bed is dry a ¼ 1; second, when the bed is filled with
water a ¼ 0. As a consequence, the void fraction a can be deduced
from the frequency f by use of the following equation:
4
a¼
Fluidisation
2a
f0
2a
f1
f0
f
2a
2a
;
ð11Þ
where f 0 ¼ f ða ¼ 0Þ and f 1 ¼ f ða ¼ 1Þ.
30
1
0
3.3. Validation
3
10
200
Capacitance probes are currently used in soil physics, generally
for water saturation measurements, but their applications are limited to quasi-static configurations, often with stratified phases.
Their operation in two phase-flow situations is therefore original
and it will now be demonstrated that this technique is indeed able
to measure the void fraction for such application.
The capacitance probe has been set-up into the CALIDE facility
to validate the void fraction measurement and to fit the parameter
a in the Lichtenecker formula. The void fraction is measured over a
volume surrounding the probe (Fig. 4). The validation method consists in comparing the results obtained with the new method to the
results that can be obtained with the weighing method used by
Tutu et al. (1984b) and Schulenberg and Müller (1987).
To measure the void fraction with the weighing method, the test
section is initially filled with water. The water level is fixed at the
upper bound of the debris bed and a spillway is located, at this elevation, on the side of the test section in order to maintain the water
level stable (Fig. 5a). Then, a constant air mass flow rate is supplied
in the test section, from the bottom, which leads to an increase of
the swollen water level and to a progressive evacuation of the water
through the spillway (Fig. 5b). The swollen water level then
decreases back to the upper bound of the debris bed, and a part of
the water initially present in the test section is recovered in a container connected to the spillway (Fig. 5c). The volume of recovered
water corresponds to the volume of air in the test section. It is
deduced from its mass m. During this phase of the test, the frequency f provided by the capacitance probe is measured by a
numerical oscilloscope. Its value is fluctuating together with the
two-phase air–water flow. The value is averaged over a time long
enough such that the mean value remains constant. A part of the
air volume stays under the grid which is supporting the debris
bed. To determine this volume, the air flow rate is suddenly stopped
(Fig. 5d). Then, there is a stratification with two layers – air and
water. The grid is fine enough such that air cannot exit by the top
through the bed. Finally, the void fraction in the bed can be deduced:
mg (Nl/min)
Fig. 3. Test phases.
The constant a is depending on the geometry of the porous space. A
theoretical justification for this mixing formula, which had hitherto
been considered as empirical, has been proposed in Zakri et al.
(1998). According to this model, the constant a is such that
1 < a < 1 and, often a 0:5 (Zakri, 1997). In our study, there are
three phases in the test section: solid (glass beads), liquid (water)
and gas (air). The solid matrix is fixed and, then, the solid fraction
is constant and equal to es ¼ 1 e. The gas and liquid fractions
are respectively equal to eg ¼ ea and el ¼ eð1 aÞ. Therefore, the
Lichtenecker formula becomes:
K a ¼ ð1 eÞK as þ eaK ag þ eð1 aÞK al :
ð7Þ
The relation between the permittivity K and the frequency f measured by the capacitance probe will now be derived. The capacitance of any capacitor is proportional to the dielectric permittivity:
C ¼ qcap K;
ð8Þ
where qcap is a parameter depending on the geometrical configuration of the capacitor, which is constant in our study (two parallel
cylindrical rods surrounded by an infinite medium). Furthermore,
the frequency is linked to the capacitance C and the inductance L
of the RLC circuit by the following relation:
2p
f ¼ pffiffiffiffiffiffi :
LC
ð9Þ
Therefore, the relation between the frequency and the void fraction
can be written as:
f
2a
¼
!2a
2p
pffiffiffiffiffiffiffiffiffiffi
eK ag eK al a þ ð1 eÞK as þ eK al :
qcap L
ð10Þ
m
8 mm
Inox beads
Measurement
volume
for P
Glass beads
hS
a ¼ ql
;
eV bed
2a
This is a linear relation between f
and a. Therefore, it can be
entirely determined with two calibration points. In our case, there
Measurement
volume
for
30 cm
Pl
Wire mesh
Fig. 4. Capacitive probe set-up.
ð12Þ
f
h
m
Fig. 5. Validation of the void fraction measurement using a weighing method.
where h is the height of the air layer under the grid, S is the area of
the test section and V bed is the volume occupied by the bed in the
test section. The uncertainty related to the weighing method has
been estimated using the Guide to the Expression of Uncertainty
in Measurement (GUM). This method is very accurate and reliable.
The uncertainty remains lower than 1% for all the tests performed
for the validation.
The validation includes series of tests on various debris bed
with different sizes and shapes (Table 3). Several air mass flow
rates were applied such that wide ranges of void fractions have
been reached, from 10% to 80%. In each test, the Lichtenecker
parameter a have been fitted. It has been concluded that the value
a ¼ 0:4 allows very good agreement between the void fraction
measurements by the capacitance probe and by the weighing
method. The comparison between both methods is presented in
Fig. 6. It demonstrates that the void fraction provided by the
capacitance probe is reliable and precise, since the standard difference between the two methods is lower than 10%. Note that the
validation tests have been performed with a zero net water flow.
Nevertheless, both phases -water and air- are locally moving due
to flow fluctuation. As said before, the frequency f is recorded
and averaged over a long time. Finally, the mean value of frequency
does not depend on the global air and water flow rate, but only the
volumic fraction of each phase, as predicted by the Lichtenecker
model. That is why the void fraction measurement can be used
with a non-zero net water flow. It has been done and the results
are presented in next section.
4. Results
Air–water flows have been generated through the prepared
beds of glass beads for various mass flow rates, as listed in Table 4.
The pressure drop and the mean void fraction have been measured.
The results are presented for each bed and compared to former
tests.
4.1. 4 mm beads debris bed
Fig. 7 presents the void fraction measurements through 4 mm
glass bead bed versus air and water filtration velocity. Each series
of points corresponds to a constant liquid velocity. The void fraction ranges from 0% to 70%. The results show that it depends
mainly on the gas velocity. The void fraction increases together
with the gas velocity, as expected. A correlation can be determined
from these data with an accuracy of 10%:
1
a ¼ 0:11Re3g ;
where Reg is the Reynolds number for the gas phase:
Reg ¼
dU g qg
:
lg ð1 eÞ
ð14Þ
The liquid velocity seems to have a negligible effect on the void
fraction.
1
1.0
Rel (-)
0.9
+10%
0.8
0.8
-10%
0.6
0.5
0.4
0.3
0.2
0.1
0,83 vg
0.7
b8 - 1
b8 - 2
b4 - 1
b4 - 2
b3 - 1
b3 - 2
c8×12 - 1
c8×12 - 2
c5×5
p4×4
0.7
-
= 0.391
= 0.391
= 0.378
= 0.379
= 0.380
= 0.399
= 0.357
= 0.364
= 0.331
= 0.327
1/3
0.6
α
0.9
αweight (-)
ð13Þ
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
0.46
0.57
0.00
0.78
1.56
2.33
3.11
4.67
6.22
7.78
29.08
43.62
58.13
72.66
109.03
145.35
vl (mm/s)
0.00
0.120
0.241
0.362
0.482
0.723
0.964
1.21
4.82
7.23
9.64
12.1
18.1
24.1
Reg (-)
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
αprobe (-)
Fig. 6. Comparison between the void fraction measurements made with the
capacitance probe and with the weighing method.
0
0.11
0.23
0.34
vg (m/s)
Fig. 7. Void fraction measurement for air–water flow through beds made with
4 mm glass beads as a function of air and water filtration velocities.
3
Rel (-)
( - dP/dz ) / ( ρl g )
2.5
2
1.5
1
0.5
0
50
100
150
200
250
0.46
0.57
0.00
0.78
1.56
2.33
3.11
4.67
6.22
7.78
29.08
43.62
58.13
72.66
109.03
145.35
vl (mm/s)
0.00
0.120
0.241
0.362
0.482
0.723
0.964
1.21
4.82
7.23
9.64
12.1
18.1
24.1
Reg (-)
of points corresponds to a constant liquid velocity. The void fraction ranges from 0% to 60%. The results are close to those obtained
with the bed made of 4 mm beads. The results show that it
depends mainly on the gas velocity. The void fraction increases
together with the gas velocity. A correlation can be determined
from these data with an accuracy of 10%:
1
a ¼ 0:085Re3g :
The liquid velocity seems to have a negligible effect on the void
fraction. It is worth noticing that the Reynolds number depends
on the particle diameter. The ratio between both correlations (13)
and (15) can be written as:
1
0
0.11
0.23
0.34
1
0:11Re3g
1
vg (m/s)
ð15Þ
0:085Re3g
¼
0:11ðd4 ð1 e8 ÞÞ3
0:085ðd8 ð1 e4 ÞÞ3
1
¼ 1:03 1:
ð16Þ
Fig. 8. Pressure drop measurement for air–water flow through beds made with
4 mm glass beads as a function of air and water filtration velocities.
Thus, the void fraction does not depend on the particle size and can
be calculated by the following equation:
Fig. 8 presents the pressure drop versus air and water filtration
velocity. The pressure drop has been normalized by the hydrostatic
pressure, such that it is equal to 1 when the water and air mass
flow rates are null. As for the void fraction measurement, each series of points corresponds to a constant liquid velocity.
For a low water injection velocity, V l < 4 mm/s, the pressure
gradient first decreases below the hydrostatic gradient with
increasing air flow rate, which is an indication of a significant force
applied by the gas phase on the liquid phase, in the upward direction. After reaching a minimum, it increases for higher air flow rate.
In the case V l ¼ 0 mm/s, it seems that the normalized pressure gradient tends to one, which is an indication that the effect of gas flow
on the pressure drop becomes negligible at high gas flow rate. This
is a striking observation which could be interpreted with the existence of at least two different flow configurations, one at low gas
velocity and another one at high gas velocity. This seems to be confirmed in Fig. 7 where there is a change of slope for V g > 0:3 m/s.
For a higher water velocity, V l > 4 mm/s, the normalized pressure gradient increases continuously with the air velocity and it
is always greater than one. This indicates that the flow configuration where the gas phase induces an upward force on the liquid
phase becomes less significant when the water flow increases. It
is an inertial effect.
a ¼ 0:83V 3g :
4.2. 8 mm balls debris bed
Fig. 9 presents the void fraction measurement through 8 mm
glass bead bed versus air and water filtration velocity. Each series
Rel (-)
0.8
0,83 vg
0.7
1/3
α
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
0.00
1.51
3.02
4.54
6.05
9.07
12.09
15.11
61.58
92.38
123.23
154.02
230.94
308.01
4.3. Comparison with former Tutu experiment
As said in the introduction, Tutu performed experiments to
study the pressure drop for air–water flow through debris beds.
These tests were carried out with a zero net water flow and provided void fraction measurements by use of a weighing method.
Three particle diameters were used to build the debris bed:
3.18 mm, 6.35 mm and 12.7 mm. These data are currently used
in the framework of modeling (Schmidt, 2006).
The results of Tutu experiments are presented together with the
results of the present work in Figs. 11 and 12.
1.6
vl (mm/s)
0.00
0.120
0.241
0.362
0.482
0.723
0.964
1.21
4.82
7.23
9.64
12.1
18.1
24.1
ð17Þ
Fig. 10 presents the pressure drop depending on air and water
filtration velocity. For low water injection velocities, V l < 12 mm/
s, the pressure gradient decreases first below the hydrostatic gradient with increasing air flow rate. After reaching a minimum, it
increases for higher air flow rates. When V l ¼ 0 mm/s, the normalized pressure gradient remains lower than 1 even for gas Reynolds
number greater than 200. This indicates that the flow configuration
where the gas phase induces an upward force on the liquid phase is
more significant for the 8 mm particles.
For higher water velocity, V l > 12 mm/s, the pressure gradient increases continuously with the air velocity and it is always
greater than 1. As will be discussed later, these qualitative
behaviors can be used to validate certain forms of the macroscale equations.
Rel (-)
1.4
( - dP/dz ) / ( ρl g )
1
0.9
1
1.2
1
0.8
0.6
0.4
0
100
Reg (-)
0
0.11
0.22
0.34
200
300
400
500
0.45
0.56
0.00
1.51
3.02
4.54
6.05
9.07
12.09
15.11
61.58
92.38
123.23
154.02
230.94
308.01
vl (mm/s)
0.00
0.120
0.241
0.362
0.482
0.723
0.964
1.21
4.82
7.23
9.64
12.1
18.1
24.1
Reg (-)
0.45
0.56
vg (m/s)
Fig. 9. Void fraction measurement for air–water flow through beds made with
8 mm glass beads as a function of air and water filtration velocities.
0
0.11
0.22
0.34
vg (m/s)
Fig. 10. Pressure drop measurement for air–water flow through beds made with
8 mm glass beads as a function of air and water filtration velocities.
1.1
3.18 mm
(Tutu et al. 1983)
1
4 mm
(Present work)
( - dP/dz ) / ( ρl g )
0.9
0.8
6.35 mm
(Tutu et al. 1983)
0.7
8 mm
(Present work)
0.6
12.7 mm
(Tutu et al. 1983)
0.5
where K l stands for the relative permeability that is a non-linear
function of a. An obvious consequence of this model is that, if
l
U l ¼ 0, one has @P
¼ ql g, which is not consistent with our experi@z
mental findings. Indeed, this model is not entirely supported by
mathematical upscaling techniques. Theoretical developments
based on the assumption of quasi-steady and quasi-static flows
and low Bond number for a Representative Elementary Volume lead
to generalized Darcy’s laws with additional crossterms accounting
for the viscous interaction between the two phases (Whitaker,
1986). Inertia terms have been added under the form of
Forchheimer terms, a popular form being the classical Ergun’s law
used in Chemical Engineering. It reads
0.4
0
0.2
0.4
0.6
vg (m/s)
Fig. 11. Comparison of the results from CALIDE facility with former Tutu experiments: normalized pressure gradient vs air velocity.
1
3.18 mm
(Tutu et al. 1983)
4 mm
(Present work)
6.35 mm
(Tutu et al. 1983)
0.1
0,83 vg
8 mm
(Present work)
1/3
12.7 mm
(Tutu et al. 1983)
0.01
0.0001
0.001
0.01
0.1
1
vg (m/s)
ð19Þ
for the l-phase, where g is called the passability and gl the relative
passability depending on a. This form is not also entirely supported
by upscaling techniques which again suggest the introduction of
viscous and inertia cross terms in addition to the Forchheimer
terms in Eq. (19) (Lasseux et al., 2008). This point is further
discussed below.
Before proceeding to this analysis, one must remember that all
the models presented above make assumptions of nearly quasisteady and quasi-static pore-scale flows. It is not proved that such
assumptions are correct in the context of nuclear safety. Indeed,
two-phase flow in highly permeable media are subject to
complex, transient pore-scale mechanisms. Theoretical models
for dynamic two-phase flows have been proposed in the literature
(Kalaydjian, 1987; Quintard and Whitaker, 1990; Hassanizadeh
and Gray, 1993; Hilfer, 1998; Panfilov and Panfilova, 2005,. . .).
This problem to date is still a rather open research domain. In this
study, we only consider the extensions of the generalized Darcy’s
law including cross-terms and inertia terms.
5.1. Pressure drop model
Fig. 12. Comparison of the results from CALIDE facility with former Tutu experiments: void fraction vs air velocity.
For low air velocity, the normalized pressure gradient decreases
together with the air velocity with the same slope regardless of the
particle diameter. However, the air velocity corresponding to
the minimum value for the pressure gradient is increasing with
the diameter.
Eq. (17) has been reported on Fig. 12. Its prediction of the void
fraction is valid, with an accuracy of 10%, for void fraction greater
than 0.3. This results is supported for all particle diameters from
3 mm to 12 mm.
Generally speaking, the results obtained in the CALIDE facility
using the capacitance probe to measure the void fraction are consistent with those of Tutu et al. (1984b). This confirms the quality
and accuracy of the new data proposed in the paper. Therefore, the
new data obtained with a non-zero water flow can be considered
as reliable.
5. Discussion
To understand what is at stake when discussing the obtained
phenomenological results, one needs to consider the type of
macro-scale momentum equations proposed in the literature. The
most widely used model corresponds to the generalized Darcy’s
law proposed by Muskat (1937). For the l-phase in 1D, we have
@Pl
l
ql g ¼ l U l ;
@z
KK l
@Pl
l
q
ql g ¼ l U l þ l U 2l
@z
KK l
ggl
ð18Þ
Several models were developed to describe the friction laws for
two-phase flows through porous medium. The models used in
severe accident codes are based on a generalization of Ergun law
from single- to two-phase flows:
@Pl
l
q
Fi
;
ql g ¼ l U l þ l U 2l @z
KK l
ggl
1a
ð20Þ
lg
qg 2 F i
@Pg
Ug þ
U þ :
qg g ¼
@z
KK g
ggg g a
ð21Þ
As discussed before, they can be divided in two groups, whether
they include (Tung and Dhir, 1988; Schulenberg and Müller,
1987) or not (Lipinski, 1982; Reed, 1982; Hu and Theofanous,
1991) a term to represent the interfacial drag F i . It is important to
note that, for stagnant water, i.e., zero net water flow, the models
which do not include a term for interfacial drag give a normalized
pressure gradient equal to 1, whatever the gas velocity (see above
discussion about Eq. (18)). For the first group, the correlations
proposed for relative permeability and passability are summarized
in Table 5.
Table 5
Pressure drop models.
Model
Kg
gg
Kl
Lipinski
a
a3
a3
a
a5
a6
Reed
Hu and Theofanous
3
3
gl
Fi
ð1 aÞ
ð1 aÞ
3
0
ð1 aÞ3
ð1 aÞ5
0
ð1 aÞ3
ð1 aÞ6
0
3
The Schulenberg and Muller model proposed the following
expression of the interfacial drag:
2
Exp
1
qK
Ug
Ul
F i ¼ 350ð1 aÞ a l ðql qg Þg
gr
a 1a
7
ð22Þ
The expressions for permeability and passability in the liquid phase
are assumed to be the same as those proposed by Reed. For the gas,
it was proposed: K g ¼ a3 and
gg ¼ 0:1a4 if 0 < a < 0:3
gg ¼ a6 if 0:3 < a < 1
ð23Þ
Lipinski
( - dP/dz ) / ( ρl g )
1.2
ð24Þ
The model of Tung and Dhir is based on visual observations of
air–water flow experiments. From that, they deduced flow patterns:
bubbly, slug and annular flows, and intermediate mixed flowing
regimes. For each regime, correlations were proposed. They are
detailed in Tung and Dhir (1988).
0.8
Reed
0.6
Hu&Theofanous
0.4
Schulenberg
0.2
Tung&Dhir
0
0
0.2
0.4
0.6
vg (m/s)
1.2
5.2. Assessment of model against experiment
Exp
two water injection velocities: V l ¼ 0 mm/s and V l ¼ 18 mm/s,
two bead sizes: 4 mm and 8 mm.
Figs. 13 and 14 present the experimental void fraction and
pressure drop measurements as a function of the gas velocity
together with the model predictions, for the case V l ¼ 0 mm/s.
( - dP/dz ) / ( ρl g )
1
Comparisons were made between the above-mentioned models
and the experimental results. In the models, the void fraction a is
obtained by an implicit resolution of Eqs. (20) and (21). Four
configurations have been investigated with:
Lipinski
0.8
Reed
0.6
Hu&Theofanous
0.4
Schulenberg
0.2
Tung&Dhir
0
0
0.2
0.4
0.6
vg (m/s)
1
0.9
Exp
0.8
Lipinski
0.7
α
0.6
Reed
0.5
Hu&Theofanous
0.4
0.3
Schulenberg
0.2
Tung&Dhir
0.1
0
0
0.2
0.4
0.6
vg (m/s)
1
0.9
Exp
0.8
Lipinski
0.7
0.6
α
Fig. 14. Pressure drop measurement for air–water flow through packed beds made
with 4 mm (left) and 8 mm (right) glass beads bed vs models for V l ¼ 0 mm=s.
Reed
0.5
Hu&Theofanous
0.4
0.3
Schulenberg
0.2
Tung&Dhir
0.1
0
0
0.2
0.4
0.6
vg (m/s)
Fig. 13. Void fraction measurement for air–water flow through packed beds made
with 4 mm (left) and 8 mm (right) glass beads vs models for V l ¼ 0 mm=s.
Given the fact that a variation of a of about 0.1 gives rise to
important variations of the relative permeability or passabilities,
which are expressed by power laws, one may consider that there
are significant differences between the estimations of the various
models. Those of Reed, Lipinski and Hu et al. give better estimates
for both particle sizes, still with differences of several per cent in
void fraction value. In addition, the model of Tung and Dhir is
the only one to catch the change of slope for the 4 mm balls,
although it overestimates the void fraction by about 0.10.
The first important observation about the normalized pressure
drop is that it is lower than 1 when increasing the gas velocity.
As discussed above, this is a behavior which is predicted by models
including the drag between the liquid and the gas. Therefore, since
this effect is not negligible, models which do not consider the interfacial drag must be discarded in most situations related to nuclear
safety analysis. Models with interaction terms (Schulenberg and
Müller, 1987; Tung and Dhir, 1988) feature a pressure gradient
decrease with increasing gas velocity in qualitative agreement
with the experimental data, but the model of Schulenberg and
Muller is the only one to predict the increase for higher gas
velocities and therefore the existence of a minimum.
Figs. 15 and 16 present the experimental results for void
fraction and pressure drop measurements together with the model
predictions for V l ¼ 18 mm/s.
The void fraction is far underestimated by the models of Lipinski
and Reed. The models of Tung and Dhir, Schulenberg and Muller, Hu
and Theofanous provide values in better agreement with the experimental data, still short of a very accurate prediction.
4
0.5
0.45
3.5
Exp
0.4
Lipinski
0.3
Reed
0.25
Hu&Theofanous
0.2
0.15
Schulenberg
0.1
( - dP/dz ) / ( ρl g )
0.35
α
Exp
3
Reed
2
1.5
Hu&Theofanous
1
Schulenberg
0.5
Tung&Dhir
0.05
Lipinski
2.5
Tung&Dhir
0
0
0
0
0.05
0.1
0.15
0.05
0.15
0.2
vg (m/s)
vg (m/s)
0.5
2
0.45
1.8
Exp
Lipinski
0.35
0.3
Reed
0.25
Hu&Theofanous
0.2
0.15
Schulenberg
Exp
1.6
( - dP/dz ) / ( ρl g )
0.4
α
0.1
0.2
1.4
Lipinski
1.2
Reed
1
Hu&Theofanous
0.8
0.6
Schulenberg
0.4
0.1
Tung&Dhir
0.05
Tung&Dhir
0.2
0
0
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
vg (m/s)
vg (m/s)
Fig. 15. Void fraction measurement for air–water flow through packed beds made
with 4 mm (left) and 8 mm (right) glass beads vs models for V l ¼ 18 mm=s.
Fig. 16. Pressure drop measurement for air–water flow through packed beds made
with 4 mm (left) and 8 mm (right) glass beads vs models for V l ¼ 18 mm=s.
All models, except Lipinski’s one, overestimate the pressure
drop. The model of Tung and Dhir, and Lipinski give fair estimation
of the pressure drop for the 8 mm beads. No model is able to propose a good estimation of the pressure drop for the 4 mm ball bed.
As a conclusion, none of the existing models is able to predict
correctly the void fraction and the pressure gradient for the 4 cases
selected from our data. This is not really surprising because those
models were only validated with data obtained for the V l ¼ 0.
New correlations have to be proposed in order to cover the whole
range of flow parameters and particle diameters which are expected
in reactor situations. From the preliminary comparisons shown in
this paper, it appears necessary to include some terms representing
the interfacial drag or, at least, the effect of the gas velocity on the
liquid phase pressure gradient, in addition to the standard relative
permeability and passability terms. This is in accordance with
theoretical suggestions based on upscaling techniques.
performed with a zero net water flow rate, and showed a very good
agreement and consistency. This gives confidence in the reliability
of the data obtained for non zero water flow rates.
The data demonstrate the strong impact of interfacial friction
phenomena and give indications on possible differences of flow
configurations, depending on the gas velocity. Finally, the classical
pressure drop models implemented in severe accident codes have
been assessed against the new data. Generally speaking, the usual
generalized of Ergun laws are not appropriate to model two-phase
flows across coarse debris beds because they cannot reproduce the
decrease of pressure drop at low gas velocity. The introduction of
an interfacial drag term provides a significant improvement in
the modeling of configurations with a zero net water flow, but
the new obtained data showed that it is not sufficient to consider
flows with Vl > 0. This is a clear indication that the phenomenology of friction between phases is not enough understood and
requires further investigations. The new set of data presented in
this paper may help deriving more relevant and accurate models.
6. Conclusions
A new method was developed for measuring the void fraction
for two-phase flows in beds of coarse particles. This technology,
based on the use of a capacitance probe, was implemented in the
CALIDE facility and validated by comparison to a weighing method.
Integral tests have been performed subsequently on single-size
debris beds made of 4 mm and 8 mm glass beads. Air–water flows
were studied, for wide and representative ranges of air and water
flow rates. The experimental campaign provided original data
including air and water velocities, pressure drop and, for the first
time, void fraction. The results were compared to former tests
Acknowledgement
The authors thanks EDF for funding part of this work.
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